Is x^4 + y^4 '>' z^4?
(1) x^2 + y^2 '>' z^2
(2) x+y '>' z
takeaway #1: when you plug numbers on a DS problem, YOUR GOAL IS TO PROVE THAT THE STATEMENT IS INSUFFICIENT.
therefore, as soon as you get a 'yes' answer, you should be TRYING to get a 'no' answer to go along with it; and, as soon as you get a 'no' answer, you should be TRYING to get a 'yes' answer to go along with it.
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statement (2)
you need to pick numbers such that x + y '>' z, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0.
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x + y > z.
fortunately, this is somewhat simple to do: just make z a big negative number.
try x = 1, y = 1, z = -100
in this case, x + y '>' z (satisfying statement two), but x^4 + y^4 is clearly less than z^4, so, NO to the prompt question.
insufficient.
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statement (1)
you need to pick numbers such that x^2 + y^2 '>' z^2, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0 (the same set of numbers we picked last time).
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x^2 + y^2 '>' z^2.
unfortunately, this isn't as easy to do as it was last time; we can't just make z a huge negative number, because z^2 would then still be a giant positive number (thwarting our efforts at obeying the criterion).
so, we have to finesse this one a bit, but the deal is still to make z as big as possible while still obeying the criterion.
let's let x and y randomly be 3 and 3.
then x^2 + y^2 = 18. we need z^2 to be less than this, but still as big as possible. so let's let z = 4 (so that z^2 = 16, which is pretty close).**
with these numbers, x^4 + y^4 = 162, which is much less than z^4 = 256. therefore, NO to the prompt question, so, insufficient.
answer = e.
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